The Problem of Missing Data

• GPA: Aligns specimens to the average shape (reference, consensus shape) - Removes non-shape information - Requires landmark correspondence among specimens, and that all landmarks must be present on all specimens

• What does one do when specimens have missing data?

Missing Data

• In some fields (e.g., anthropology), missing data is pervasive
  • Specimens are incomplete, or lack structures entirely

Dealing with Missing Data: Delete Specimens

• Simplest solution is to remove specimens with missing data

Dealing with Missing Data: Delete Specimens

• Simplest solution is to remove specimens with missing data

• Reduces sample size
• No information from deleted specimens, which could be important (e.g., a rare species)

• NOT an optimal solution!

Dealing with Missing Data: Delete Landmarks

• Another option is to delete landmarks that are missing from some specimens

Dealing with Missing Data: Delete Landmarks

• Another option is to delete landmarks that are missing from some specimens

• Retains original sample size
• No information from deleted landmarks, which could be biologically important

• NOT an optimal solution!

Estimate Missing Data

• An alternative is to estimate missing data in some intelligent manner
• Goal is to generate ‘complete’ specimens by estimating missing landmarks
  • Called data imputation in the statistical literature
• Use some reasonable procedure to predict missing landmark locations
• Several approaches possible

See Gunz et al. (2009). J. Hum. Evol.

1: Exploiting Symmetry

• Use biological symmetry to estimate missing landmarks
• Two approaches:
  • 1: Mirror-image (reflect) portion of structure to the other side
  • 2: Create \(\small{2}^{nd}\) full specimen via reflection (and relabeled)
• Locations of missing landmarks estimated from location in reflected portion

1: Exploiting Symmetry

• Use biological symmetry to estimate missing landmarks
• Two approaches:
  • 1: Mirror-image (reflect) portion of structure to the other side
  • 2: Create \(\small{2}^{nd}\) full specimen via reflection (and relabeled)
• Locations of missing landmarks estimated from location in reflected portion

1: Exploiting Symmetry

• Use biological symmetry to estimate missing landmarks
• Two approaches:
  • 1: Mirror-image (reflect) portion of structure to the other side
  • 2: Create \(\small{2}^{nd}\) full specimen via reflection (and relabeled)
• Locations of missing landmarks estimated from location in reflected portion

Note: the ‘reflectMissingLandmarks’ function in StereoMorph may be used for symmetry-based estimation of missing landmarks

Exploiting Symmetry: Example

• Here we have the lizard head dataset
• Let’s take one specimen (in red) and eliminate some landmarks

Exploiting Symmetry: Example

• Here we have the lizard head dataset
• Let’s take one specimen (in red) and eliminate some landmarks

• Let’s delete some landmarks then estimate by exploiting symmetry

Exploiting Symmetry: Test Procedure

• 1: Obtain specimen, delete landmarks
• 2: Estimate landmarks via symmetry
• 3: Calculate \(\small{D}_{Proc}\) between original and estimated

Exploiting Symmetry: Test Procedure

• 1: Obtain specimen, delete landmarks
• 2: Estimate landmarks via symmetry
• 3: Calculate \(\small{D}_{Proc}\) between original and estimated

\(\small{D}_{Proc}= 0.009\). Pretty good!

Exploiting Symmetry

• Advantages
  • Exploits spatial relationships of anatomy within a specimen
  • Leverages ‘pseudoreplication’ of symmetric points
• Disadvantages
  • Not all objects are symmetric
  • Studies of asymmetry are challenged, because by definition only the symmetric portion of shape is used in the reconstruction

• Symmetry methods can be useful but not are not a general solution (limited to symmetric structures)

2: Mean Substitution

• Use landmarks in reference to estimate missing landmarks
  • 1: Superimpose all complete specimens
  • 2: Obtain reference (average)
  • 3: Replace missing landmarks with values from reference

see Arbour and Brown. (2014). Methods. Ecol. Evol.

Mean Substitution: Example

• Here we have some (simulated) fish data:

• Let’s create missing landmarks for several specimens and predict via mean substitution

Mean Substitution: Example 1

• Let’s take one specimen (in red) and eliminate some landmarks

• Now let’s delete some landmarks then estimate by mean substitution

Mean Substitution: Example 1 (Cont.)

Test Procedure

• 1: Obtain specimen, delete landmarks
• 2: Estimate landmarks from mean specimen
• 3: Calculate \(\small{D}_{Proc}\) between original and estimated

\(\small{D}_{Proc_{Ref-Orig}} = 0.26\) \(\small{D}_{Proc_{Ref-Est}} = 0.23\)

\(\small{D}_{Proc_{Orig-Est}} = 0.13\) Not good at all!

Mean Substitution: Example 2

• Or let’s take a different specimen (in red) and eliminate some landmarks

• Now let’s delete some landmarks then estimate by mean substitution

Mean Substitution: Example 2 (Cont.)

Test Procedure

• 1: Obtain specimen, delete landmarks
• 2: Estimate landmarks from mean specimen
• 3: Calculate \(\small{D}_{Proc}\) between original and estimated

\(\small{D}_{Proc_{Ref-Orig}} = 0.15\) \(\small{D}_{Proc_{Ref-Est}} = 0.13\)

\(\small{D}_{Proc_{Orig-Est}} = 0.08\) Not good at all!

Mean Substitution

• What’s going on here?
  • Mean substitution does not account for systematic variation (e.g., allometry)
  • If shape covaries with some factor (e.g., size!), estimates will be off
  • Will over- or under-estimate landmark locations relative to the specimen

Mean Substitution

• What’s going on here?
  • Mean substitution does not account for systematic variation (e.g., allometry)
  • If shape covaries with some factor (e.g., size!), estimates will be off
  • Will over- or under-estimate landmark locations relative to the specimen

• Mean substitution should not be used!

3: TPS Interpolation

• Use thin-plate spline to estimate location of missing landmarks

3: TPS Interpolation

• Use thin-plate spline to estimate location of missing landmarks

• Procedure
  • 1: Identify common landmarks in both reference and target
  • 2: Calculate TPS interpolation
  • 3: Locations of missing landmarks in target estimated from their location in reference, filtered through TPS

Bookstein et al. (1999). New. Anat.; Gunz et al. (2009). J. Hum. Evol.

TPS Interpolation: Concept

• Procedure
  • 1: Identify common landmarks in both reference and target
  • 2: Calculate TPS interpolation
  • 3: Locations of missing landmarks in target estimated from their location in reference, filtered through TPS

Bookstein et al. (1999). New. Anat.; Gunz et al. (2009). J. Hum. Evol.

TPS Interpolation: Example 1

\(\small{D}_{Proc_{Ref-Orig}} = 0.26\) \(\small{D}_{Proc_{Ref-Est}} = 0.27\)

\(\small{D}_{Proc_{Orig-Est}} = 0.003\) MUCH Better!

TPS Interpolation: Example 2

\(\small{D}_{Proc_{Ref-Orig}} = 0.15\) \(\small{D}_{Proc_{Ref-Est}} = 0.15\)

\(\small{D}_{Proc_{Orig-Est}} = 0.011\) MUCH Better!

TPS Interpolation

Advantages - Exploits spatial relationships of anatomy within a specimen

Disadvantages - Less accurate if many landmarks in a region missing (common with fossils) - Does not leverage additional covariation information in sample

• TPS interpolation is very useful, but may be improved upon

4: Regression Interpolation

• Use covariation between landmarks to estimate locations

• Procedure

  • 1: Superimpose all complete specimens
  • 2:Regress landmarks with missing values against complete specimens
  • 3: Use post-hoc prediction on regression for missing landmarks
  • 4: Predicted values serve as missing landmark locations

Note: Regression scores of PLS typically used as \(\small{p>n}\)

See Gunz et al. (2009). J. Hum. Evol.; Arbour and Brown (2014). Methods. Ecol. Evol.

Regression Interpolation: Example 1

\(\small{D}_{Proc_{Ref-Orig}} = 0.26\) \(\small{D}_{Proc_{Ref-Est}} = 0.27\)

\(\small{D}_{Proc_{Orig-Est}} = 0.030\) Pretty good!

Regression Interpolation: Example 2

\(\small{D}_{Proc_{Ref-Orig}} = 0.15\) \(\small{D}_{Proc_{Ref-Est}} = 0.15\)

\(\small{D}_{Proc_{Orig-Est}} = 0.011\) Even Better!

Regression Interpolation

Advantages - Exploits spatial relationships of anatomy within a specimen - Leverages covariation between anatomical landmarks - Leverages covariation within a sample

Disadvantages - May be less accurate when small samples are examined

NOTE: Estimation may be further improved by considering within-sample variation (e.g., use specimens within a species)

• Regression interpolation is VERY useful, but is it universally better?

Methods Comparisons

• Few systematic comparisons among methods exist, but those that do imply that regression estimation is generally preferred, followed by TPS interpolation

• HIgher error with \(\small{\uparrow}\) missing landmarks and \(\small{\uparrow}\) specimens containing missing data

• Regression method generally preferred

Flow of Computations

• GMM Workflow should be augmented to account for missing landmarks
  • 1: Digitize data & read into geomorph
  • 2: Estimate missing landmarks
  • 3: GPA + projection
  • 4: Statistical analyses and visualization

Missing Data: Conclusions

• Missing data has been a major challenge to GM analyses

• Deleting specimens or landmarks ignores information

• Morphometric-based estimation

  • Exploit symmetry in data
  • Use TPS interpolation

• ‘Classical’ statistical approaches extended to morphometrics

  • Regression incorporates covariation in sample

• Regression approach appears most robust